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fCopyright © 2013 Tech Science Press CMC, vol.1, no.1, pp.1-23, 20131

The Cell Method: an Enriched Description of Physics2

Starting from the Algebraic Formulation3

E. Ferretti14

Abstract: In several recent papers studying the Cell Method (CM), which is a5

numerical method based on a truly algebraic formulation, it has been shown that6

numerical modeling in physics can be achieved even without starting from differ-7

ential equations, by using a direct algebraic formulation. In the present paper, our8

focus will be above all on highlighting some of the theoretical features of this al-9

gebraic formulation to show that the CM is not simply a new numerical method10

among many others, but a powerful numerical instrument that can be used to avoid11

spurious solutions in computational physics.12

Keywords: Algebraic Formulation, Differential Formulation, Cell Method, Spu-13

rious Solutions, Nonlocality.14

1 Introduction15

From the onset of differential calculus, over three centuries ago [Newton (1687)],16

we have become accustomed to providing a differential formulation to each exper-17

imental law. Infinitesimal analysis has without doubt played a major role in the18

mathematical treatment of physics in the past, and will continue to do so in the19

future, but we must also be aware that, in using it, several important aspects of20

the phenomenon being described, such as its geometrical and topological features21

[Tonti (in press)], remain hidden. Moreover, applying the limit process introduces22

some limitations as regularity conditions must be imposed on the field variables.23

These regularity conditions, in particular those concerning differentiability, are the24

price we pay for using a formalism that is both very advanced and easy to manipu-25

late.26

Since the arrival of computers, differential equations have been discretized using27

one of various discretization methods (the finite element method FEM, the bound-28

1 DICAM – Department of Civil, Environmental and Materials Engineering, Scuola di Ingegneria eArchitettura, Alma Mater Studiorum, Università di Bologna, Viale Risorgimento 2, 40136 (BO),ITALY.


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f2 Copyright © 2013 Tech Science Press CMC, vol.1, no.1, pp.1-23, 2013

ary element method BEM, the finite volume method FVM, the finite difference29

method FDM, etc.), since the numerical solution, which is no longer an exact so-30

lution, cannot be achieved for the most general case if a system of algebraic phys-31

ical laws is not provided. Nevertheless, the very need to discretize the differential32

equations, in order to achieve a numerical solution, gives rise to the question of33

whether or not it is possible to formulate physical laws in an algebraic manner di-34

rectly, through a direct algebraic formulation. We will see, in this paper, that this is35

possible and that a truly algebraic numerical method, the Cell Method (CM [Tonti36

(2001)]), besides being physically more appealing, also allows us to avoid some of37

the typical numerical problems of differential formulation, since it would appear38

that many numerical problems are purely the result of the generally consolidated39

custom of formulating the problem in differential form.40

2 Basics of algebraic formulation41

The starting point in algebraic formulation [Tonti (1998); Tonti (2001); Freschi, Gi-42

accone, and Repetto (2008); Alotto, Freschi, and Repetto (2010); Alotto, Freschi,43

Repetto, and Rosso (2013)] is that only a few physical variables arise directly as44

functions of points and instants. Most of them are obtained by performing densities45

and rates on variables related to extended space elements and time intervals.46

We will call:47

• global variable in space, a variable that is not the line, surface or volume48

density of another variable;49

• global variable in time, a variable that is not the rate of another variable.50

2.1 Global variables and their features51

Global variables are essential to the philosophy of the Cell Method, since, by using52

these variables, it is possible to obtain an algebraic formulation directly and, what53

is most important, the global variables involved in obtaining the formulation do not54

have to be differentiable functions. Therefore, by using the limit process on the55

mean densities and rates of the global variables, we can obtain the traditional field56

functions of the differential formulation.57

The main difference between the two formulations – algebraic and differential – lies58

precisely in the fact that the limit process is used in the latter. In effect, since cal-59

culating the densities and rates of the domain variables is based on the assumption60

that global variables are continuous and differentiable, the range of applicability61

of differential formulation is restricted to regions without material discontinuities62

or concentrated sources, while that of the algebraic formulation is not restricted63


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fThe Cell Method 3

to such regions [Ferretti (2004a,b); Ferretti, Casadio, and Di Leo (2008); Ferretti64

(2013)].65

Among the possible classifications of physical variables, the one we will adopt in66

this work makes the distinction between:67

• configuration variables, which describe the field configuration;68

• source variables, which describe the field sources.69

Displacements in solid mechanics, velocity in fluid dynamics, electric potential in70

electrostatics and temperature in thermal conduction are all examples of config-71

uration variables, while forces in solid mechanics and fluid dynamics, masses in72

geodesy, electric charges in electrostatics, electric currents in magnetostatics and73

heat in thermal conduction are instead examples of source variables.74

The equations used to relate the configuration variables of the same physical theory75

to each other and the source variables of the same physical theory to each other76

are known as topological equations, while those that relate configuration to source77

variables, of the same physical theory, are known as constitutive equations.78

Since each physical phenomenon occurs in space, and space has a multi-dimensional79

geometrical structure, the physical variables themselves have a multi-dimensional80

geometrical content. As a consequence, each of the global physical variables is81

associated with one of the four space elements: point (P), line (L), surface (S) or82

volume (V). The association between physical variables and space elements in di-83

mensions 0, 1, 2 and 3 is ignored in differential formulation, while it is emphasized84

in the Cell Method, where it becomes the corner stone for building the relative85

governing equations. As we will see better later, the fact that the Cell Method86

takes account of the association between physical variables and space elements is87

also the main reason why it does not present the spurious solutions of differen-88

tial formulation [Pimprikar, Teresa, Roy, Vasu, and Rajan (2013); Qian, Han, and89

Atluri (2013); Kakuda, Nagashima, Hayashi, Obara, Toyotani, Katsurada, Higuchi,90

and Matsuda (2012); Kakuda, Obara, Toyotani, Meguro, and Furuichi (2012); Cai,91

Tian, and Atluri (2011); Chang (2011); Dong and Atluri (2011); Liu (2012); Jarak92

and Sori\’c, (2011); Liu and Atluri (2011); Liu, Dai, and Atluri (2011); Wu and93

Chang (2011); Cai, Paik, and Atluri (2010); Liu, Hong, and Atluri, (2010); Soares,94

D. Jr. (2010); Yeih, Liu, Kuo, and Atluri (2010); Zhu, Cai, Paik, and Atluri (2010)].95

We will use algebraic topology notations to describe the four space elements P, L,96

S and V. In algebraic topology, it is usual to consider cell-complexes, and to denote97

the vertexes as 0-cells, the edges as 1-cells, the surfaces as 2-cells and the volumes98

as 3-cells (Fig. 1).99


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Figure 1: The four space elements in algebraic topology

Figure 2: Association between space elements and variables in continuum mechan-ics

Figure 3: Faces of a p-cell of degree 1, 2 and 3.


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fThe Cell Method 5

Figure 4: Cofaces of a p-cell of degree 0, 1 and 2.

In continuum mechanics, volume forces, which are source variables, are associated100

with 3-cells, since their geometrical referents are volumes (Fig. 2). Analogously,101

for surface forces, which are source variables, the geometrical referents (the sur-102

faces) are 2-cells; for strains, which are configuration variables, the geometrical103

referents (the lines) are 1-cells; and for displacements, which are configuration104

variables, the geometrical referents (the points) are 0-cells.105

Algebraic topology also involves the notions of face and coface. If we consider a106

cell-complex made of p-cells of degree 0, 1, 2 and 3, the (p−1)-cells that bound a107

given p-cell are the faces of the p-cell (Fig. 3). The set of faces of a p-cell defines108

the boundary of the p-cell.109

The (p+1)-cells that have a given p-cell as a common face are the cofaces of that110

p-cell (Fig. 4). The set of cofaces of a p-cell defines the coboundary of the p-cell.111

2.2 How the Cell Method works112

The Cell Method has often been compared to the Direct or Physical Approach, ini-113

tially used in the Finite Element Method [Huebner (1975); Livesley (1983); Fenner114

(1996)], or to the Finite Volume Method and the Finite Difference Method. In par-115

ticular, the Cell Method may seem very similar to the vertex-based scheme of the116

FVM [Mavripilis (1995)]. However, on deeper analysis of the similarities and dif-117

ferences between the CM and other discrete methods, the CM is shown to be based118

on a new philosophy, where, for the moment, the CM is seen as the only truly alge-119

braic method. In effect, the key point to bear in mind in building a truly algebraic120

formulation is that all operators must be discrete and use of the limit process must121

be avoided at each level of the formulation. The direct or physical approach is not122

suited to this, since it starts from the point-wise conservation equations of differen-123

tial formulation (Fig. 5) and, for differential formulation, there is the need for field124

functions, which depend on the point position and the instant value. If the field125

functions are not described directly in terms of point position and instant values,126


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they can be obtained by calculating the densities and rates of the global variables,127

which are domain variables and depend on the point position and the instant value,128

and also on line extensions, areas, volumes and time intervals.129

Figure 5: Building an algebraic formulation through the Direct or Physical Ap-proach, the Finite Volume Method, the Finite Difference Method and the CellMethod

The space distribution of the point-wise field functions requires the introduction of130

coordinate systems (Fig. 5), whose purpose is to create a correspondence between131

the points of the space and the numbers, that is, their coordinates. This allows us to132

describe geometry through mathematics.133

The algebraic formulation can be derived from the differential formulation through134

an integration process (Fig. 5) that is needed because, while in differential formula-135

tion geometry must be eliminated from the physical laws, in the numerical solution136

geometry is essential.137

The Finite Volume Method and the Finite Difference Method are also based on a138

differential formulation (Fig. 5). The Cell Method, on the contrary, uses global139

variables and balance equations in a global form. As a consequence, the relative140

governing equations are expressed directly in algebraic form.141


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fThe Cell Method 7

The CM uses cell-complexes (Fig. 5), which are the generalization of the coordi-142

nate systems to the algebraic formulation. The CM cell-complexes are not actually143

the result of a domain discretization, a process needed in numerical analysis, as in144

the case of the FEM. They are required in algebraic formulation, since global vari-145

ables are associated not only with points, as for differential formulation, but also146

with lines, surfaces and volumes. In the CM, global variables are described directly147

as they will be associated with the related space elements of the cell-complexes.148

Consequently, in the algebraic formulation of physics, cell complexes have the149

same role that coordinate systems have in differential formulation. Physical no-150

tions are therefore translated into mathematical notions through the intermediation151

of topology and geometry.152

The geometrical structure of space is very rich in algebraic formulation. It is pos-153

sible, for example, to define an inner orientation for the elements in dimension 0,154

1, 2 and 3 of a cell-complex (Fig. 5), which we call the primal cell-complex. Then,155

by considering the planes that are equidistant from the primal nodes, we can define156

a second cell complex (Fig. 5), called the dual cell-complex, which turns out to be157

provided with an outer orientation.158

Moreover, if we consider a time axis and subdivide a given time interval into many159

adjacent small time intervals (Fig. 5), we have a primal cell complex in time. In160

order to build the dual cell complex in time, we will consider the middle instant of161

each time interval. The result is that, similarly for the space elements, the primal162

and dual time elements are also provided with inner and outer orientation.163

We will now see how the space elements of the primal and dual cell-complexes164

are strictly associated with global variables. For the example, in Fig. 6, on a two-165

dimensional domain, once a mesh has been introduced, it is natural to associate the166

primal nodes with the displacements of the primal nodes and the total load over167

an area surrounding the primal nodes, which is an area of the dual cell complex.168

It follows that the displacements, which are configuration variables, are computed169

on objects of the primal mesh, while the loads, which are source variables, are170

computed on objects of the dual complex.171

This result is general, independently of the kind of configuration or source variable,172

the shape of the domain, or the physical theory involved. In effect, for each set of173

primal nodes and for each given physical theory, the source variables are always as-174

sociated with the elements of the dual cell-complex and the configuration variables175

are always associated with the elements of the primal cell-complex (Fig. 7).176

The association of physical variables with the elements of a cell complex and its177

dual was first introduced by Okada and Onodera (1951) and Branin (1966). In the178

CM, the strong coupling between physical variables and oriented space elements179


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Figure 6: Association between global variables and elements of the two cell com-plexes

Figure 7: Association between global variables and space elements of the primaland dual cell complexes, in different physical theories

becomes the key to giving a direct discrete formulation to physical laws.180

Moreover, the existence of an underlying structure, common to different physical181

theories, is mainly responsible for the structural similarities presented in physical182

theories, commonly called “analogies”. Today, we are able to explain these analo-183

gies in the light of the association between the global variables and the four space184

elements, since the homologous global variables of two physical theories are those185

associated to the same space element. In other words, the analogies between phys-186

ical theories arise from the geometrical structure of the global variables and not187


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fThe Cell Method 9

from the similarity of the equations that relate variables to each other in different188

physical theories [Tonti (in press)].189

3 The structure of the governing equations in the CM190

3.1 Nonlocality in algebraic and differential formulations191

For the nodes of a dual cell-complex in plane domains, we can choose, for example,192

the barycenters of a primal cell-complex made of triangles, a simplicial primal193

mesh (Fig. 8).194

Since even in plane domains the primal mesh has thickness, which is a unit thick-195

ness, the dual nodes are not in the same plane as the primal nodes and the two196

meshes are shifted along the thickness (Fig. 9).197

This is similar to the relative position of staggered elements, a commonly used198

mathematical expedient to avoid spurious solutions in physics. In particular, in199

solid mechanics, imbricate elements [Bažant, Belytschko, and Chang (1984); Be-200

lytschko, Bažant, Hyun, and Chang (1986)] are examples of staggered elements201

used for regularizing material instability in strain-softening materials, both in one-202

and two-dimensional domains (Fig. 10).203

As it is now clear that global physical variables are naturally associated with the204

space elements of the primal and dual cell-complexes, it therefore follows that stag-205

gering is not only a mathematical expedient used to regularize the solution [Stevens206

and Power (2010)], but is also necessary in physics to take account of the associa-207

tion between physical variables and oriented space elements.208

In solid mechanics, staggering is used to provide the differential formulation with209

nonlocal properties, when modeling heterogeneous materials. It is now the com-210

mon opinion that the classical local continuum concept, where stress at a given211

point depends only on the deformation and temperature history at that precise point,212

cannot adequately describe damage in heterogeneous materials by means of differ-213

ential formulation, particularly when size-effect is involved. Indeed, modeling the214

size-effect is impossible in the context of classical plasticity, both in problems in-215

volving strain-softening [Duhem (1893); Krumhansl (1965); Rogula (1965); Erin-216

gen (1966); Kunin (1966); Kröner (1968)] and in those with no strain-softening at217

all.218

The first criticisms of the local approach date back to the 1960s [Krumhansl (1965),219

Rogula (1965), Eringen (1966), Kunin (1966) and Kröner (1968)] and are based220

on the microstructure of matter. In effect, all materials are characterized by mi-221

crostructural details, with size ranging over several orders of magnitude [Bažant222

and Jirásek (2002)]. They cannot, therefore, be broken down into a set of infinites-223


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Figure 8: Primal and dual cell complexes in plane domains

Figure 9: Staggering of the primal and dual cell complexes along the thickness

Figure 10: Imbricate elements in one- and two-dimnsional domains


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fThe Cell Method 11

imal volumes, each of which can be described independently. Consequently, the224

idea was advanced that heterogeneous materials should be modeled properly by225

some kind of nonlocal continuum [Duhem (1893)], in which the stress at a certain226

point is a function of the strain distribution over a certain representative volume227

centered at that point [Bažant and Chang (1984)]. This idea led to models where228

the classical continuum description is improved by introducing an internal length229

parameter into the constitutive laws.230

According to the mathematical definition of nonlocality in the narrow sense, givenby Rogula, the operator A in the abstract form of the fundamental equations of anyphysical theory:

Au = f , (1)

is called local when, if u(x) = v(x) for all x in a neighborhood of point x0, then231

Au(x) = Av(x). Bažant and Jirásek (2002) and Ferretti (2005) pointed out that the232

differential operators satisfy this condition, because the derivatives of any arbitrary233

order do not change if the differentiated function only changes outside the small234

neighborhood of the point where the derivatives are taken, and, consequently, the235

differential operators are local. It follows that any formulation using differential236

operators is intrinsically local. That is, differential formulation is not adequate for237

describing nonlocal effects.238

Figure 11: Losing and reintroducing metrics in the differential formulation

In the light of the former discussion on the geometrical content of global variables,239

we can now provide an alternative interpretation of nonlocality. As we have dis-240

cussed previously, the reason why differential operators are local in nature lies in241


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the use of the limit process. The density-finding process is carried out with the242

intention of formulating the field laws in an exact form. However, differential243

formulation can be solved only for very simple geometries and under particular244

boundary conditions (Fig. 11). Moreover, with the global variables being reduced245

to point and instant variables, we can no longer describe more than 0-dimensional246

effects, that is, the nonlocal effects. Metrics must be reintroduced a-posteriori in247

the discretization process, if we want to model nonlocality (Fig. 11).248

We may now ask where the length scale is to be reintroduced. In nonlocal ap-249

proaches, a length scale is incorporated into the constitutive laws, but there is no250

evidence that this choice is the only one, or even the most appealing from a physi-251

cal point of view. On the contrary, we have seen that the physical global variables252

themselves have a multi-dimensional geometrical content. It therefore seems that253

dimensional scales and nonlocal effects are directly associated with global vari-254

ables, and nonlocality seems to be a property of global variables, not a preroga-255

tive of constitutive laws. Consequently, reintroducing or preserving nonlocality in256

governing equations is physically more correct than reintroducing nonlocality into257

constitutive equations. When speaking of reintroduction, we are dealing with dif-258

ferential formulation, while, when speaking of preservation, we are dealing with259

algebraic formulation. The difference is not negligible, since, in order to rein-260

troduce a length scale, a suitable approach must be developed, while, in order to261

preserve the length scales, it is sufficient to avoid the limit process and, by using an262

algebraic approach, a nonlocal formulation is automatically obtained.263

Besides this, due to the structure of the discrete operators, even the topological264

equations are provided with nonlocal properties in algebraic formulation. The rea-265

son for this lies in the relationship between the discrete p-forms of different de-266

grees.267

3.2 The discrete p-forms (cochains)268

A physical variable φ associated with one set of p-cells of a cell-complex defines269

a discrete p-form (or a discrete form of degree p). The potential of a vector field,270

line integral of a vector, flux and mass content are discrete forms of degree 0, 1, 2271

and 3, respectively (Tab. 1).272

Table 1: Examples of discrete p-formsVariable Potential of a

vector fieldLine integral ofa vector

Flux Mass content

Evaluated on 0-cells (points) 1-cells (lines) 2-cells (surfaces) 3-cells (volumes)Discrete p-form discrete 0-form discrete 1-form discrete 2-form discrete 3-form


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fThe Cell Method 13

The discrete p-forms generalize the notion of field functions, because, in a discrete273

p-form Φ[S], we associate the value of a physical variable to the space elements274

of degree p, while the field functions f (P) always associate the value of a physical275

variable to the points of the domain. As a consequence, Φ[S] is a set function,276

while f (P) is a point function.277

The notion of discrete form is the discrete version of the exterior differential form,278

a mathematical formalism that has the great merit of highlighting the geometri-279

cal background of physical variables, something ignored by differential calculus,280

providing a description that is independent of the coordinate system used. Never-281

theless, this formalism uses field variables instead of global variables and, for this282

reason, it must use the notion of derivative.283

3.3 The coboundary process and its implications on nonlocality284

The coboundary process on a discrete p-form is a process that generates a discrete285

(p+1)-form. It is worth noting that balance, circuital equations, and equations form-286

ing differences can be expressed by the coboundary process performed on discrete287

p-forms of degree 2, 1, 0, respectively. Thus, the coboundary process plays a key288

role in physics.289

This process is analogous, in an algebraic setting, to the exterior differentiation on290

exterior differential forms and leads to discrete operators, whose elements are the291

incidence numbers, equal to 0, +1, or -1. In particular, the incidence number of a292

p-cell with a (p−1)-cell is equal to:293

• 0, if the (p−1)-cell is not on the boundary of the p-cell;294

• +1, if the (p−1)-cell is on the boundary of the p-cell and the orientations of295

the p-cell and (p−1)-cell are compatible (Fig. 12);296

• −1, if the (p−1)-cell is on the boundary of the p-cell and the orientations of297

the p-cell and (p−1)-cell are not compatible (Fig. 13).298

In a three-dimensional space, we can define three incidence matrices:299

• G: matrix of the incidence numbers between 1-cells and 0-cells;300

• C: matrix of the incidence numbers between 2-cells and 1-cells;301

• D: matrix of the incidence numbers between 3-cells and 2-cells.302

The incidence matrices G, C, and D are the discrete versions of the differential303

operators “grad”, “curl”, and “div”, respectively.304

The coboundary process on a discrete p-form, Φp, is performed in two steps:305


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Figure 12: Incidence numbers equal to +1

Figure 13: Incidence numbers equal to -1

1. For each p-cell: we assign the value φpn , evaluated on the nth p-cell, to each306

coface (of degree p+1) of the nth p-cell, with the plus or minus sign accord-307

ing to the mutual incidence number;308

2. For each (p+1)-cell: we sum the values φp1,2,...,h coming from all the faces (of309

degreep) of the boundary.310

At the end of the process, we obtain a discrete (p+1)-form.311

The consequence of this two-step process, when we enforce balance in the CM,312

which is a coboundary process on a discrete p-form of degree 2, the flux, described313

as follows:314

• first step (on each 2-cell): transfer the fluxes associated with all 2-cells to315

their cofaces, each multiplied by the relative incidence number;316


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fThe Cell Method 15

• second step (on each 3-cell): perform the algebraic sum of the fluxes coming317

from the first step;318

is that the global source variables involved in the balance of one 3-cell are also319

involved in the balance of all the surrounding 3-cells. This happens since each 3-320

cell is common to more than one 2-cell. Consequently, the balance at a given 3-cell321

does not depend on the current values, or previous history, of the global source322

variables at that 3-cell only, but on the current values, and previous history, taken323

by the global source variables in all the surrounding 3-cells. This gives nonlocal324

properties to the balance equations of the algebraic formulation, while the balance325

equations of the differential formulation are local.326

Achieving nonlocality in the CM balance equations is very important, as it enriches327

the description of physics given by the CM, compared to the descriptions given by328

any other numerical method, even those known as discrete methods. This enrich-329

ment follows on from the structure of the coboundary process given to the balance330

equations and from more than one 2-cell sharing the same 3-cell. That is, it is331

a consequence of the structure of the balance equations, and there is no need to332

modify the balance equations in any way to provide them with nonlocal properties.333

Moreover, coboundary processes on discrete p-forms defined on configuration vari-334

ables generate discrete (p+1)-forms defined on configuration variables, and cobound-335

ary processes on discrete p-forms defined on source variables generate discrete336

(p+1)-forms defined on source variables. Thus, the coboundary process creates337

a relationship between different degrees of discrete forms of the same kind of338

variable. This means that all the topological equations are coboundary processes339

[Ferretti (in press)], and therefore, even enforcing compatibility in the CM is a340

coboundary process. As a consequence, even the Kinematic equations are pro-341

vided with nonlocal properties in the CM, while the Kinematic equations of the342

differential formulation are local. We can therefore conclude that all the governing343

equations of the algebraic formulation are nonlocal: constitutive relationships are344

nonlocal due to the staggering between the elements of the primal cell complex, on345

which we compute configuration variables, and the dual cell complex, on which we346

compute source variables, while the balance and Kinematic equations are nonlocal,347

since they are the results of coboundary processes. This means that obtaining a348

nonlocal formulation by using discrete operators is possible, besides being physi-349

cally appealing. The new nonlocal formulation is desirable from a numerical point350

of view, since the numerical solution is reached sooner using discrete operators351

than with differential operators.352


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3.4 Implementation of the equations353

The CM equations are implemented in the same manner as for FEM. The linearinterpolation of the CM for solid mechanics in two-dimensional domains was pro-vided in Ferretti (2003a), with a Kinematic equation for each primal cell:

εεε = BBBuuu, (2)

uuu =[

wk vk wi vi w j v j]T

, (3)

BBB =1

∆i jk

∆y ji 0 ∆yk j 0 ∆yik 00 ∆xi j 0 ∆x jk 0 ∆xki∆xi j ∆y ji ∆x jk ∆yk j ∆xki ∆yik

. (4)

a constitutive law:

σσσ = DDDεεε. (5)

and an equilibrium equation for each dual cell (Fig. 14):

∑∑∑ j QQQk j +FFFk = 0, (6)

QQQ = NNNσσσ = NNNDDDεεε = (((NNNDDDBBB)))uuu, (7)

NNN =

[Sx 0 Sy

0 Sy Sx

], (8)

SSS = RRRLLL. (9)

The linear system of equations can be written in the form:

FFF = KKKUUU , (10)

where FFF and UUU are the force and displacement vectors, respectively, and KKK is anal-354

ogous to the FEM stiffness matrix, which is symmetric and defined as positive for355

properly constrained systems.356

The CM was also implemented for solid mechanics with quadratic interpolation of357

the displacements in two-dimensional domains [Cosmi (2000)] and three-dimensional358

domains [Pani and Taddei (2013)].359

Lastly, Zovatto (2001) proposed a meshfree approach of the CM, including for360

three-dimensional domains. Meshfree and meshless approaches are very useful361

in problems of fracture mechanics where the crack being studied is simulated as362

a discontinuity of the displacement field. In effect, crack geometry updating and363

remeshing on the whole domain is a very expensive process from a computational364


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fThe Cell Method 17

Figure 14: Primal and dual meshes for CM analysis in two-dimensional domains

point of view. Some of the most recent achievements for the CM meshless ap-365

proach can be found in Pani and Taddei (2013) and Taddei, Pani, Zovatto, Tonti,366

and Viceconti (2008).367

As far as the convergence rate is concerned, it has been shown [Tonti (2001)] that368

this depends on the choice of dual polygons. The most convenient choice, giving a369

convergence rate equal to four [Cosmi (2000)], is to use Gauss points to build the370

dual polygons.371

4 Conclusions372

In the present paper, we have discussed where the Cell Method stands in respect373

to the discussion on local or nonlocal descriptions of the continuum for modeling374

heterogeneous brittle materials. We have found that the CM can provide a direct375

nonlocal description of the continuum, without requiring any sort of enrichment to376

the constitutive laws, by means of length scales, as is usually the case for nonlocal377

approaches in solid mechanics. In particular, we can state that the CM does not pro-378

vide an enriched continuum description only as far as the constitutive relationships379

are concerned. It enriches all the governing equations involved in the physical the-380

ory in a very simple manner, by simply taking account of the association between381

global variables and extended space elements. The consequence is that, by using382

the CM, there is no need to recover nonlocality a-posteriori, as for differential for-383

mulation. Nonlocality is – we could say – intrinsic to algebraic formulation and is384

the result of using global instead of field variables, something that distinguishes the385

cell method from any other numerical method, at the moment.386


Proo


As already discussed in Ferretti [2005], it is worth noting that, in the first theories387

of nonlocal elasticity developed by Eringen and Edelen [Eringen (1966); Edelen et388

al. (1971); Eringen (1972); Eringen and Edelen (1972)], nonlocality was a property389

of the elastic problem in its complex, and not solely of its constitutive relationships.390

In other words, in these nonlocality theories, there was already the idea that non-391

locality is a property of the governing equations. Nevertheless, this idea was not392

developed further, since the theories of nonlocal elasticity were too complicated to393

be calibrated and verified experimentally, let alone to be applied to any real prob-394

lems [Bažant and Jirásek (2002)]. Treating only the stress-strain relationships as395

nonlocal, while the equilibrium and kinematic equations and their corresponding396

boundary conditions retain their standard form, was something needed later [Erin-397

gen and Kim (1974); Eringen et al. (1977)], to provide a practical formulation of398

these early theories. Consequently, incorporating the length scale into the constitu-399

tive relationships only is the practical simplification of a more general theory and400

has no evident justification from a physical point of view. In this sense, we can401

state that the cell method provides a physically more appealing nonlocal formu-402

lation when compared to nonlocal differential approaches. Ferretti provides sev-403

eral numerical results [Ferretti (2012), Ferretti (2005), Ferretti (2004c) and Ferretti404

(2003b)] showing how the CM, together with a new local constitutive law for het-405

erogeneous brittle materials – the effective law [Ferretti (2004d); Ferretti (2004e);406

Ferretti (2004f); Ferretti and Di Leo (2008)] – actually offers a nonlocal descrip-407

tion for solid mechanics, allowing us to model the size- and shape-effects, which is408

impossible in local differential approaches.409

A properly formulated enriched classical continuum for the differential formulation410

has a regularizing effect when modeling strain-softening materials, because it acts411

as a localization limiter, so that the boundary value problem is once again well-412

posed. Boundary value problems with strain-softening constitutive models that are413

ill-posed are just some of the many examples of spurious solutions for differential414

formulation. Since the CM is a multidisciplinary method, the intrinsic nonlocality415

of the CM governing equations means that we can assume that, using the CM,416

spurious solutions can be avoided whenever they appear in differential formulation,417

independently of the physical theory involved. This is the really strong point for418

the CM, setting it apart from all other numerical methods, for the moment.419

In conclusion, the Cell Method enriches the computational description of physics420

by using notions of algebraic topology, as well as of mathematics. The result is421

twofold:422

• The Cell Method provides the physical laws in an algebraic manner, directly,423

avoiding the use of any discretization procedure.424


Proo

fThe Cell Method 19

• The Cell Method is tantamount to abandoning the principle of local action,425

therefore of avoiding the spurious solutions of differential formulation.426

References427

Alotto, P.; Freschi, F.; Repetto, M. (2010): Multiphysics problems via the cell428

method: The role of Tonti diagrams. IEEE Transactions on Magnetics, vol. 46, no.429

8, pp. 2959–2962.430

Alotto, P.; Freschi, F.; Repetto, M.; Rosso, C. (2013): The Cell Method for Elec-431

trical Engineering and Multiphysics Problems - An Introduction. Lecture Notes in432

Electrical Engineering, Springer.433

Bažant, Z. P.; Chang, T.P. (1984): Is Strain-Softening Mathematically Admissi-434

ble? Proc., 5th Engineering Mechanics Division, vol. 2, pp. 1377–1380.435

Bažant, Z. P.; Belytschko, T. B.; Chang, T.P. (1984): Continuum Model for Strain436

Softening. J. Eng. Mech., vol. 110, no. 12, pp. 1666–1692.437

Bažant, Z. P.; Jirásek, M. (2002): Nonlocal Integral Formulations of Plasticity438

and Damage: Survey of Progress. J. Eng. Mech., vol. 128, no. 11, pp. 1119–1149.439

Belytschko, T.; Bažant, Z. P.; Hyun, Y.-W.; Chang, T.-P. (1986): Strain Soften-440

ing Materials and Finite-Element Solutions. Comput. Struct., vol. 23, pp. 163–180.441

Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P. (1996): Mesh-442

less Methods: an Overview and Recent Developments. Computer Methods Appl.443

Mech Engrg., vol. 139, pp 3–47.444

Branin, F. H. Jr. (1966): The Algebraic Topological Basis for Network Analogies445

and the Vector Calculus. Proc., Symp. on Generalized Networks, Brooklyn Polit.,446

pp. 453–487.447

Cai, Y.C.; Paik, J.K.; Atluri, S.N. (2010): Locking-free thick-thin rod/beam ele-448

ment for large deformation analyses of space-frame structures, based on the reiss-449

ner variational principle and a von karman type nonlinear theory. CMES: Comput.450

Model. Eng. Sci., vol. 58, no. 1, pp. 75-108.451

Cai, Y.C.; Tian, L.G.; Atluri, S.N. (2011): A simple locking-free discrete shear452

triangular plate element. CMES: Comput. Model. Eng. Sci., vol. 77, no. 3-4, pp.453

221-238.454

Chang C.-W. (2011): A New Quasi-Boundary Scheme for Three-Dimensional455

Backward Heat Conduction Problems. CMC: Comput. Mater. Con., vol. 24,456

no. 3, pp. 209–238.457

Cosmi, F. (2000): Applicazione del Metodo delle Celle con Approssimazione458

Quadratica. Proc., AIAS 2000, Lucca, Italy, pp. 131–140.459

Dong, L.; Atluri, S.N. (2011): A simple procedure to develop efficient & stable460


Proo


hybrid/mixed elements, and Voronoi cell finite elements for macro- & microme-461

chanics. CMC: Comput. Mater. Con., vol. 24, no. 1, pp. 61–104.462

Duhem, P. (1893): Le Potentiel Thermodynamique et la Pression Hydrostatique.463

Ann. Sci. Ecole Norm. S., vol. 10, pp. 183–230.464

Edelen, D.G.B.; Green, A. E.; Laws, N. (1971): Nonlocal Continuum Mechanics.465

Arch. Ration. Mech. Anal., vol. 43, pp. 36–44.466

Eringen, A.C. (1966): A Unified Theory of Thermomechanical Materials. Int. J467

Eng. Sci., vol. 4, pp. 179–202.468

Eringen, A.C. (1972): Linear Theory of Nonlocal Elasticity and Dispersion of469

Plane Waves. Int. J Eng. Sci., vol. 10, pp. 425–435.470

Eringen, A.C.; Edelen, D. G. B. (1972): On Nonlocal Elasticity. Int. J Eng. Sci.,471

vol. 10, pp. 233–248.472

Eringen, A.C.; Kim, B. S. (1974): Stress Concentration at the Tip of a Crack.473

Mech. Res. Commun., vol. 1, pp. 233–237.474

Eringen, A.C.; Speziale, C. G.; Kim, B. S. (1977): Crack-Tip Problem in Nonlo-475

cal Elasticity. J. Mech. Phys. Solids, vol. 25, pp. 339–355.476

Fenner, R.T. (1996): Finite Element Methods for Engineers, Imperial College477

Press, London.478

Ferretti, E. (2003a): Crack Propagation Modeling by Remeshing using the Cell479

Method (CM). CMES: Comput. Model. Eng. Sci., vol. 4, no. 1, pp. 51–72.480

Ferretti, E. (2003b): Modeling of Compressive Tests on FRP Wrapped Concrete481

Cylinders trough a Novel Triaxial Concrete Constitutive Law. SITA: Scientific Is-482

rael – Technological Advantages, vol. 5, pp. 20–43.483

Ferretti, E. (2004a): Crack-Path Analysis for Brittle and Non-Brittle Cracks: a484

Cell Method Approach. CMES: Comput. Model. Eng. Sci., vol. 6, no. 3, pp.485

227-244.486

Ferretti, E. (2004b): A Cell Method (CM) Code for Modeling the Pullout Test487

Step-Wise. CMES: Comput. Model. Eng. Sci., vol. 6, no. 5, pp. 453–476.488

Ferretti, E. (2004c): A Discrete Nonlocal Formulation using Local Constitutive489

Laws. Int. J. Fracture, vol. 130, no. 3, pp. L175–L182.490

Ferretti, E. (2004d): A Discussion of Strain-Softening in Concrete. Int. J. Fracture491

(Letters section), vol. 126, no. 1, pp. L3–L10.492

Ferretti, E. (2004e): Experimental Procedure for Verifying Strain-Softening in493

Concrete. Int. J. Fracture (Letters section), vol. 126, no. 2, pp. L27–L34.494

Ferretti, E. (2004f): On Poisson’s Ratio and Volumetric Strain in Concrete. Int. J.495

Fracture (Letters section), vol. 126, no. 3, pp. L49–L55.496


Proo

fThe Cell Method 21

Ferretti, E. (2005): A Local Strictly Nondecreasing Material Law for Modeling497

Softening and Size-Effect: a Discrete Approach. CMES: Comput. Model. Eng.498

Sci., vol. 9, no. 1, pp. 19–48.499

Ferretti, E. (2012): Shape-Effect in the Effective Law of Plain and Rubberized500

Concrete. CMC: Comput. Mater. Con., vol. 30, no. 3, pp. 237–284.501

Ferretti, E. (2013): A Cell Method Stress Analysis in Thin Floor Tiles Subjected502

to Temperature Variation. Submitted to CMC: Comput. Mater. Con.503

Ferretti, E. (in press): The Cell Method: a Purely Algebraic Computational Method504

in Physics and Engineering Science, Momentum Press.505

Ferretti, E.; Casadio, E.; Di Leo, A. (2008): Masonry Walls under Shear Test: a506

CM Modeling. CMES: Comput. Model. Eng. Sci., vol. 30, no. 3, pp 163–190.507

Ferretti, E.; Di Leo, A. (2008): Cracking and Creep Role in Displacement at508

Constant Load: Concrete Solids in Compression. CMC: Comput. Mater. Con.,509

vol. 7, no. 2, pp. 59–80.510

Freschi, F.; Giaccone, L.; Repetto, M. (2008): Educational value of the algebraic511

numerical methods in electromagnetism. COMPEL - The International Journal for512

Computation and Mathematics in Electrical and Electronic Engineering, vol. 27,513

no. 6, pp. 1343–1357.514

Okada, S.; Onodera, R. (1951): Algebraification of Field Laws of Physics by515

Poincar Process. Bull. of Yamagata University – Natural Sciences, vol. 1, no. 4,516

pp. 79–86.517

Huebner, K. H. (1975): The Finite Element Method for Engineers, Wiley.518

Kakuda, K.; Nagashima, T.; Hayashi, Y.; Obara, S.; Toyotani, J.; Katsurada,519

N.; Higuchi, S.; Matsuda, S. (2012): Particle-based Fluid Flow Simulations on520

GPGPU Using CUDA. CMES: Comput. Model. Eng. Sci., vol. 88, no. 1, pp521

17–28.522

Jarak, T.; Sori\’c, J. (2011): On Shear Locking in MLPG Solid-Shell Approach.523

CMES: Comput. Model. Eng. Sci., vol. 81, no. 2, pp 157–194.524

Kakuda, K.; Obara, S.; Toyotani, J.; Meguro, M.; Furuichi, M. (2012): Fluid525

Flow Simulation Using Particle Method and Its Physics-based Computer Graphics.526

CMES: Comput. Model. Eng. Sci., vol. 83, no. 1, pp 57–72.527

Kröner, E. (1968): Elasticity Theory of Materials with Long-Range Cohesive528

Forces. Int. J. Solids Struct., vol. 3, pp. 731–742.529

Krumhansl, J. A. (1965): Generalized Continuum Field Representation for Lattice530

Vibrations. In Lattice dynamics, R. F. Wallis ed., Pergamon, London, pp. 627–634.531

Kunin, I. A. (1966): Theory of Elasticity with Spatial Dispersion. Prikl. Mat.532


Proo


Mekh. (in Russian), vol. 30, pp. 866.533

Liu, C.-S. (2012): Optimally Generalized Regularization Methods for Solving Lin-534

ear Inverse Problems. CMC: Comput. Mater. Con., vol. 29, no. 2, pp. 103–128.535

Liu, C.-S.; Atluri, S.N. (2011): An iterative method using an optimal descent536

vector, for solving an Ill-conditioned system Bx = b, better and faster than the537

conjugate gradient method. CMES: Comput. Model. Eng. Sci., vol. 80, no. 3-4, pp538

275–298.539

Liu, C.-S.; Dai, H.-H.; Atluri, S.N. (2011): Iterative Solution of a System of Non-540

linear Algebraic Equations F(x)=0, Using dot x=λ [αR+βP] or dot x=λ [αF+βP∗]541

R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P∗ Normal to542

F. CMES: Comput. Model. Eng. Sci., vol. 81, no. 3, pp 335–363.543

Liu, C.-S.; Hong, H.-K.; Atluri, S.N. (2010): Novel algorithms based on the544

conjugate gradient method for inverting ill-conditioned matrices, and a new regu-545

larization method to solve ill-posed linear systems. CMES: Comput. Model. Eng.546

Sci., vol. 60, no. 3, pp 279–308.547

Livesley, R. K. (1983): Finite Elements, an Introduction for Engineers, Cambridge548

University Press.549

Mavripilis, D. J. (1995): Multigrid Techniques for Unstructured Meshes. Lecture,550

Series 1995-02, Computational Fluid Dynamics, Von Karman Institute of Fluid551

Dynamics.552

Newton, I. (1687): Philosophiae Naturalis Principia Mathematica.553

Pani, M.; Taddei, F. (2013): The Cell Method: Quadratic Interpolation with554

Tetraedra for 3D Scalar Fields. To appear in CMES: Comput. Model. Eng. Sci.555

Pimprikar, N.; Teresa, J.; Roy, D.; Vasu, RM.; Rajan. K. (2013): An approx-556

imately H1-optimal Petrov-Galerkin meshfree method: application to computation557

of scattered light for optical tomography. CMES: Comput. Model. Eng. Sci., vol.558

92, no. 1, pp. 33–61.559

Qian, Z.Y.; Han, Z.D.; Atluri, S.N. (2013): A fast regularized boundary integral560

method for practical acoustic problems. CMES: Comput. Model. Eng. Sci., vol.561

91, no. 6, pp. 463–484.562

Rogula, D. (1965): Influence of Spatial Acoustic Dispersion on Dynamical Prop-563

erties of Dislocations. I. Bulletin de l’Académie Polonaise des Sciences, Séries des564

Sciences Techniques, vol. 13, pp. 337–343.565

Soares, D. Jr. (2010): A Time-Domain Meshless Local Petrov-Galerkin Formula-566

tion for the Dynamic Analysis of Nonlinear Porous Media. CMES: Comput. Model.567

Eng. Sci., vol. 66, no. 3, pp. 227–248.568

Stevens, D.; Power, W. (2010): A Scalable Meshless Formulation Based on RBF569


Proo

fThe Cell Method 23

Hermitian Interpolation for 3D Nonlinear Heat Conduction Problems. CMES:570

Comput. Model. Eng. Sci., vol. 55, no. 2, pp. 111–146.571

Taddei, F.; Pani, M.; Zovatto, L.; Tonti, E.; Viceconti, M. (2008): A New Mesh-572

less Approach for Subject-Specific Strain Prediction in Long Bones: Evaluation of573

Accuracy. Clinical Biomechanics, vol. 23, no.9, pp. 1192–1199.574

Tonti, E. (1998): Algebraic Topology and Computational Electromagnetism. Fourth575

International Worksop on the Electric and Magnetic Field: from Numerical Models576

to industrial Applications, Marseille, 284-294.577

Tonti, E. (2001): A Direct Discrete Formulation of Field Laws: the Cell Method.578

CMES: Comput. Model. Eng. Sci., vol. 2, no. 2, pp. 237–258.579

Tonti, E. (in press): The Mathematical Structure of Classical and Relativistic Phys-580

ical Theories, Springer.581

Wu, J.-Y.; Chang, C.-W. (2011): A Differential Quadrature Method for Multi-582

Dimensional Inverse Heat Conduction Problem of Heat Source. CMC: Comput.583

Mater. Con., vol. 25, no. 3, pp. 215–238.584

Yeih, W.; Liu, C.-S.; Kuo, C.-L.; Atluri, S.N. (2010): On solving the direct/inverse585

cauchy problems of laplace equation in a multiply connected domain, using the586

generalized multiple-source-point boundary-collocation Trefftz method & charac-587

teristic lengths. CMC: Comput. Mater. Con., vol. 17, no. 3, pp. 275–302.588

Zhu, H.H.; Cai, Y.C.; Paik, J.K.; Atluri, S.N. (2010): Locking-free thick-thin589

rod/beam element based on a von Karman type nonlinear theory in rotated reference590

frames for large deformation analyses of space-frame structures. CMES: Comput.591

Model. Eng. Sci., vol. 57, no. 2, pp. 175–204.592

Zovatto, L. (2001): Nuovi Orizzonti per il Metodo delle Celle: Proposta per un593

Approccio Meshless. Proc., AIMETA – GIMC (in Italian), Taormina, Italy, pp.594

1–10.595


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